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On \(C^*\)-algebras related to constrained representations of a free group. (English) Zbl 1254.46060

Summary: We consider representations of the free group \(F_{2}\) on two generators for which the norm of the sum of the generators and their inverses is bounded by some number \(\mu \in [0, 4]\). These \(\mu \)-constrained representations determine a \(C^*\)-algebra \(A_{\mu }\) for each \(\mu \in [0, 4]\). If \(\mu = 4\), this gives the full group \(C^*\)-algebra of \(F _{2}\). We prove that these \(C^*\)-algebras form a continuous bundle of \(C^*\)-algebras over [0, 4] and evaluate their \(K\)-groups.

MSC:

46L05 General theory of \(C^*\)-algebras
22D25 \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations
46L80 \(K\)-theory and operator algebras (including cyclic theory)
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References:

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